Multiplicities in the Length Spectrum and Growth Rate of Salem Numbers

Abstract

We prove that mean multiplicities in the length spectrum of a non-compact arithmetic hyperbolic orbifold of dimension \(n \geqslant 4\) have exponential growth rate

$$\begin{aligned} \langle g(L) \rangle \geqslant c \frac{e^{([n/2] - 1)L}}{L^{1 + \delta _{5, 7}(n) }}, \end{aligned}$$

extending the analogous result for even dimensions of Belolipetsky, Lalín, Murillo and Thompson. Our proof is based on the study of (square-rootable) Salem numbers. As a counterpart, we also prove an asymptotic formula for the distribution of square-rootable Salem numbers by adapting the argument of Götze and Gusakova. It shows that one can not obtain a better estimate on mean multiplicities using our approach.